Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-4z^3 + 8z^2 + 192z}{-7z^3 + 98z^2 - 336z}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-4z(z^2 - 2z - 48)} {-7z(z^2 - 14z + 48)} $ $ t = \dfrac{4z}{7z} \cdot \dfrac{z^2 - 2z - 48}{z^2 - 14z + 48} $ Simplify: $ t = \dfrac{4}{7} \cdot \dfrac{z^2 - 2z - 48}{z^2 - 14z + 48}$ Since we are dividing by $z$ , we must remember that $z \neq 0$ Next factor the numerator and denominator. $ t = \dfrac{4}{7} \cdot \dfrac{(z - 8)(z + 6)}{(z - 8)(z - 6)}$ Assuming $z \neq 8$ , we can cancel the $z - 8$ $ t = \dfrac{4}{7} \cdot \dfrac{z + 6}{z - 6}$ Therefore: $ t = \dfrac{ 4(z + 6)}{ 7(z - 6)}$, $z \neq 8$, $z \neq 0$